by John Allen Paulos
Every time I read or watch anything about the election I hear some variant of the phrase “margin of error.” My mathematically attuned ears perk up, but usually it’s just a slightly pretentious way of saying the election is very close or else that it’s not very close. Schmargin of error might be a better name for metaphorical uses of the phrase.
To be fair, the phrase is often supplemented with precise numbers (plus or minus 1.5%, for example) that purport to quantify exactly how tight the race is (or isn’t). Unfortunately these numbers are not as reliable as they might seem. The problem is that an enabling condition for this precision is that a random sample of voters be polled and the larger it is, the better.
A few technical remarks on the meaning of the margin of error in the next three paragraphs, which can be skimmed or skipped.
The basic qualitative idea: If we imagine many random samples of voters being taken, the sample percentages supporting a candidate will vary from sample to sample, of course, but these sample percentages will naturally cluster around the true percentage, P, of voters supporting the candidate in the whole population.
Importantly, this clustering of the sample probabilities can be described more quantitatively if we’re dealing with random samples of voters. In fact, if we assume that p is the percentage of voters supporting candidate A in a random sample and n is the number of voters in the sample, then we can get a good estimate of P, which is what we really want to know.
Specifically, the interval ranging from -2√[p(1-p)/n] to +2√[p(1-p)/n] will encompass, P, the percentage of voters in the whole population supporting candidate A, about 95% of the time.
Half of the above interval, which will vary a bit depending on p in the particular sample taken, is the margin of error. Since n appears in the denominator, the larger the sample is, the narrower the interval encompassing P. Read more »

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